Axiom ax-mulass 7928

Description: Multiplication of complex numbers is associative. Axiom for real and complex numbers, justified by Theorem axmulass 7886. Proofs should normally use mulass 7956 instead. (New usage is discouraged.) (Contributed by NM, 22-Nov-1994.)

Assertion

Ref	Expression
ax-mulass	⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 · 𝐵) · 𝐶) = (𝐴 · (𝐵 · 𝐶)))

Detailed syntax breakdown of Axiom ax-mulass

Step	Hyp	Ref	Expression
1		cA	. . . 4 class 𝐴
2		cc 7823	. . . 4 class ℂ
3	1, 2	wcel 2158	. . 3 wff 𝐴 ∈ ℂ
4		cB	. . . 4 class 𝐵
5	4, 2	wcel 2158	. . 3 wff 𝐵 ∈ ℂ
6		cC	. . . 4 class 𝐶
7	6, 2	wcel 2158	. . 3 wff 𝐶 ∈ ℂ
8	3, 5, 7	w3a 979	. 2 wff (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ)
9		cmul 7830	. . . . 5 class ·
10	1, 4, 9	co 5888	. . . 4 class (𝐴 · 𝐵)
11	10, 6, 9	co 5888	. . 3 class ((𝐴 · 𝐵) · 𝐶)
12	4, 6, 9	co 5888	. . . 4 class (𝐵 · 𝐶)
13	1, 12, 9	co 5888	. . 3 class (𝐴 · (𝐵 · 𝐶))
14	11, 13	wceq 1363	. 2 wff ((𝐴 · 𝐵) · 𝐶) = (𝐴 · (𝐵 · 𝐶))
15	8, 14	wi 4	1 wff ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 · 𝐵) · 𝐶) = (𝐴 · (𝐵 · 𝐶)))

This axiom is referenced by:  mulass  7956